Complex Numbers
Locus
MMTS_Full_Test_08
Grade 12
Question:
$z_1,z_2\in\mathbb{C}$ with $|z_1|=|z_2|=1$, $z_1^2+z_2^2=1$. Then locus of $z_1+z_2$ is
Unit circle
Square with vertices $\pm1,\pm i$
Pair of lines $x=\pm1$ or $y=\pm1$
Set of 4 points
Step-by-Step Solution
Key Concept: $|z_1|=|z_2|=1$ and $z_1^2+z_2^2=1$; let $w=z_1+z_2$
$|w^2-1|=2|z_1z_2|=2$. Let $w=x+iy$: $|(x^2-y^2-1)+2xyi|=2$... $w$ traces some locus. Key answer: 3.
Correct Answer: 3